This falls short of the original goal: While your method shows that for every point, you can find a tiling large enough that covers it, the original question was to find a single tiling covering everything.

Finding progressively larger, finite tilings is not the same as having a single infinite tiling, just like finding larger and larger natural numbers is not the same as having a single number larger than all natural numbers (which wouldn't be a natural number).

König's lemma implies that for tilings both statements are in fact equivalent.